2.2. LAPLACE’S EQUATION 29
c. Local estimates for harmonic functions. Now we employ the mean-
value formulas to derive careful estimates on the various partial derivatives
of a harmonic function. The precise structure of these estimates will be
needed below, when we prove analyticity.
THEOREM 7 (Estimates on derivatives). Assume u is harmonic in U.
Then
(18)
|Dαu(x0)|
≤
Ck
rn+k
u
L1(B(x0,r))
for each ball B(x0,r) ⊂ U and each multiindex α of order |α| = k.
Here
(19) C0 =
1
α(n)
, Ck =
(2n+1nk)k
α(n)
(k = 1,... ).
Proof. 1. We establish (18), (19) by induction on k, the case k = 0 being
immediate from the mean-value formula (16). For k = 1, we note upon
differentiating Laplace’s equation that uxi (i = 1,...,n) is harmonic. Con-
sequently
(20)
|uxi (x0)| = −
B(x0,r/2)
uxi dx
=
2n
α(n)rn
∂B(x0,r/2)
uνi dS
≤
2n
r
u
L∞(∂B(x0,
r
2
)).
Now if x ∈
∂B(x0,r/2), then B(x, r/2) ⊂ B(x0,r) ⊂ U, and so
|u(x)| ≤
1
α(n)
2
r
n
u
L1(B(x0,r))
by (18), (19) for k = 0. Combining the inequalities above, we deduce
|Dαu(x0)|
≤
2n+1n
α(n)
1
rn+1
u
L1(B(x0,r))
if |α| = 1. This verifies (18), (19) for k = 1.
2. Assume now k ≥ 2 and (18), (19) are valid for all balls in U and each
multiindex of order less than or equal to k − 1. Fix B(x0,r) ⊂ U and let α
